# Quantum AG code[1]

## Description

A Galois-qudit CSS code constructed using two linear AG codes.

## Rate

Quantum AG codes [1] can be asymptotically good. There exist three such families [2–4] that admit a diagonal transversal gate at the third level of the Clifford hierarchy.

## Magic

By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the Clifford hierarchy and attains a zero magic-state yield parameter, \(\gamma = 0\) [2]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see [5,7][6; Sec. 5.3]. Two other such asymptotically good families exist [3,4], admitting a different diagonal gate at the third level of the Clifford hierarchy.

## Encoding

## Transversal Gates

There exist three asymptotically good code families [2–4] that admit a diagonal transversal gate at the third level of the Clifford hierarchy.

## Parent

- Galois-qudit CSS code — Quantum AG codes can be realized in the CSS code construction [9].

## Children

- Quantum Goppa code
- Galois-qudit RS code — Galois-qudit RS codes are constructed via the CSS construction from RS codes, which are evaluation AG codes.

## Cousins

- Algebraic-geometry (AG) code
- Triorthogonal code — By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the Clifford hierarchy and attains a zero magic-state yield parameter, \(\gamma = 0\) [2]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see [5,7][6; Sec. 5.3]. Two other such asymptotically good families exist [3,4], admitting a different diagonal gate at the third level of the Clifford hierarchy.
- Tsfasman-Vladut-Zink (TVZ) code — The AG codes used in an asymptotically good construction of quantum AG codes with non-Clifford transversal gates [3] are those of the TVZ codes.
- Galois-qudit GRS code — Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction from GRS codes, which are evaluation AG codes.

## References

- [1]
- R. Matsumoto, “Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes”, IEEE Transactions on Information Theory 48, 2122 (2002) arXiv:quant-ph/0107129 DOI
- [2]
- A. Wills, M.-H. Hsieh, and H. Yamasaki, “Constant-Overhead Magic State Distillation”, (2024) arXiv:2408.07764
- [3]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
- [4]
- Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
- [5]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [6]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [7]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [8]
- A. Ashikhmin and E. Knill, “Nonbinary Quantum Stabilizer Codes”, (2000) arXiv:quant-ph/0005008
- [9]
- A. Niehage, “Nonbinary Quantum Goppa Codes Exceeding the Quantum Gilbert-Varshamov Bound”, Quantum Information Processing 6, 143 (2006) DOI

## Page edit log

- Victor V. Albert (2024-08-23) — most recent

## Cite as:

“Quantum AG code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_ag